[Solution Manual For Applied Nonlinear Control Slotine.zip
Everardo Laboy
The well-orchestrated use of distilled experience, domain-specific knowledge, and well-informed trade-off decisions is imperative if we are to design effective architectures for complex software-intensive systems. In particular, designing modern self-adaptive systems requires intricate decision-making over a remarkably complex problem space and a vast array of solution mechanisms. Nowadays, a large number of approaches tackle the issue of endowing software systems with self-adaptive behavior from different perspectives and under diverse assumptions, making it harder for architects to make judicious decisions about design alternatives and quality attributes trade-offs. It has currently been claimed that search-based software design approaches may improve the quality of resulting artifacts and the productivity of design processes, as a consequence of promoting a more comprehensive and systematic representation of design knowledge and preventing design bias and false intuition. To the best of our knowledge, no empirical studies have been performed to provide sound evidence of such claim in the self-adaptive systems domain.
This study contributes to reveal empirical evidence on the benefits of search-based approaches when designing self-adaptive systems architectures. The results presented herein increase our belief that the systematic representation of distilled design knowledge and the adoption of search-based design approaches indeed lead to improved architectures.
Modern software-intensive systems are becoming increasingly complex and the fulfillment of requirements for performance, flexibility, dependability, and energy-efficiency in uncertain and dynamic environments is still a quite challenging task (Huebscher and McCann 2008). Elastic data storage services, energy-aware mobile systems, self-tuning databases, and reconfigurable network services are some of the application domains in which self-adaptive mechanisms play a paramount role (Patikirikorala et al. 2012). Such scenarios are usually characterized by incomplete knowledge about user requirements, workloads, and available resources. As a consequence, committing to a particular solution in design time may yield suboptimal architectures, which easily degrade the service when conditions deviate from those previously defined. Establishing the foundations that enable the systematic design, development, and evolution of systems with self-management capabilities has been the focus of many research efforts in areas such as self-adaptive systems, autonomic computing, and artificial intelligence (Salehie and Tahvildari 2009).
The remainder of this paper is organized as follows. Section 2 presents the foundations of SA systems and feedback control. Section 2 describes the automated software architecture design approach proposed by us and adopted as one of the experiment treatments. Section 2 presents an overview of the experiment. Section 2 explains the experiment objects, the hypotheses being investigated, the adopted measurement approach, and the experiment design. In Section 2, we analyze and discuss the experiment results. Threats to validity are identified in Section 2 and related work is discussed in Section 2. Finally, conclusions and venues for future work are presented in Section 2.
Being the resulting system stable, the remaining SASO properties can be investigated. As depicted in Figure 2a, the smaller the steady-state error e ss (difference between the reference input and measured output), the more accurate is the resulting system. The settling time K s is the time elapsed from the change in input to when the measured output is within some variation range (usually 2%) of its steady-state value. Finally, the maximum overshoot M p is the normalized maximum amount by which the system output exceeds its steady-state value.
Feedback control properties (a) and step response of systems with different controllers (b). Properties such as settling time (K s), steady-state error (e ss), and overshoot (M p) are commonly affected by the control law and tuning techniques chosen for the feedback controller managing the target system.
The systematic design of control architectures which exhibit intentionally chosen values of accuracy, settling time and overshoot is imperative if we are to conceive effective self-adaptive systems. Disregarding such an aspect may lead to over/under provisioning of resources (due to inaccurate convergence), violations of service level agreements (due to slower responses), or excessive use of resources during transient response as a consequence of large overshoots. Figure 2b depicts the step response (dynamics exhibited by the target system when the reference input changes from 0 to 1) for systems with different controllers. The controller 3 presents an ideal response, with no overshoot, high accuracy, and small settling time.
A large body of knowledge regarding control laws and methods for designing controllers is currently available (Patikirikorala et al. 2012). Currently adopted control-theoretic approaches for endowing systems with self-adaptation capabilities include the use of PID control, state-space models, MIMO (Multiple-Input Multiple-Output) control, gain scheduling, self-tuning regulators, fluid flow analysis, and fuzzy control (Tilbury et al. 2004). As a consequence, designing effective architectures for SA systems requires architects become familiar with the intricacies of both the problem space (so that accurate and realistic self-adaptation requirements can be elicited) and solution space (in order to adopt the most effective adaptation strategy/mechanism for the problem at hand). That involves deciding on self-adaptation goals; system and environment monitoring mechanisms; measurement noises and uncertainties; unanticipated/unforeseen adaptations; diverse control robustness degrees; change enacting mechanisms; and adaptation temporal predictability, just to mention a few (Andersson et al. 2009; Brun et al. 2009; Patikirikorala et al. 2012).
Each design dimension entails a set of variation points, representing alternative solutions for such a concern (e.g.: leader-followers or half-sync/half-async; for the concurrency strategy dimension).
The set of all design dimension instances generated by ds, when evaluated in M, provides the underlying infrastructure of our search-based approach for automating the architecture design process.
Therefore, a candidate architecture (a location in such n-dimensional space) is formed by the initial model extended with the merge of all architectural extensions provided by all involved variation points.
As a consequence of such an infrastructure, huge design spaces may easily be spawned even for small input models, motivating the adoption of meta-heuristics and multi-objective optimization approaches. The number of different candidate vectors in \(\mathcal D_\textit asds\) (including those resulting in invalid architectures) is given by:
Once a concrete design space is defined, architects can submit initial models to manual design space exploration or rely on the multi-objective optimization engine we provide (design space usage stage). The domain-independent optimization engine we provide handles all required steps to forge candidate architectures for a given set of design space locations, evaluate their quality regarding the attributes defined for the design space, and find out a set of Pareto-optimal architectures.
The aforementioned infrastructure provides the underpinnings of our SA systems design approach. We have specified a particular DuSE instance (SA:DuSE) that captures the most prominent degrees of freedom and quality attributes when designing adaptation loops based on feedback control (Tilbury et al. 2004). Currently, SA:DuSE yields architectural extensions regarding seven different control laws (Tilbury et al. 2004) (Proportional, Integral, Proportional-Integral, Proportional-Derivative, Proportional-Integral-Derivative, Static State Feedback, and Dynamic State Feedback), seven empirical tuning approaches (Wang 2005) (four Chien-Hrones-Reswick variations, Ziegler-Nichols, Cohen-Coon, and Linear Quadratic Regulator), five mechanisms for control adaptation (Landau et al. 2011) (fixed gain, gain scheduling, model-reference, model-identification, and reconfiguring control), and six different multiple loops arrangements (Weyns et al. 2010) (no cooperation, information sharing, coordinated control, regional planning, master/slave, and hierarchical).
In addition, four quality metrics (objective functions) evaluate the resulting architectures regarding the average settling time, average overshoot, control overhead, and control robustness. The first three metrics are intended to be minimized, while the last one is intended to be maximized. It is well-known from studies (Tilbury et al. 2004) in control theory field that settling time and average overshoot represent conflicting control goals (as presented in Figure 2b). The same has been observed for control overhead and control robustness metrics. One of our research goals was to investigate to which extent the proposed SA:DuSE design space captures such trade-offs when automating the design of SA systems architectures (as discussed below). Moreover, the architectural decision space produced by SA:DuSE exhibited 8,643,600 candidate vectors for an input model with two controllable ports. For models with four controllable ports, such number rapidly increases to 7.4711821e13, further motivating the need for effective search-based approaches. Further information about the SA:DuSE design dimensions, its corresponding variation points, and the adopted quality metrics may be found in (Andrade and de Arajo Macdo 2013a).
The outcome of our approach provides useful insights and supports the self-adaptive systems architect in several aspects. First, we observe that architectures exhibiting short average settling times are quite rare in the final population, making it harder for novice architects to find out such effective solutions by manually scouring the design space or by performing random searches. Second, the outcome reveals pronounced trade-offs between two pairs of quality attributes: i) average settling time and average overshoot (first column, second row); and ii) control robustness and control overhead (third column, fourth row). The Pareto-fronts for such combinations are smooth, providing alternative solutions regarding the fulfillment of such quality attributes. No significant trade-offs have been found in other quality metric pairs. Third, the rigorous identification of Pareto-optimal solutions prevents novice architects from adopting those combinations of control law, tuning technique, and control adaptation mechanism that lead to inferior architectures. Finally, the metric values presented by solutions in the Pareto-front allow for the early analysis of the dynamics exhibited by real prototypes implementing such architectures.
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